Theorem carden2b 10036

Description: If two sets are equinumerous, then they have equal cardinalities. (This assertion and carden2a 10035 are meant to replace carden 10620 in ZF without AC.) (Contributed by Mario Carneiro, 9-Jan-2013.) (Proof shortened by Mario Carneiro, 27-Apr-2015.)

Assertion

Ref	Expression
carden2b	⊢ (𝐴 ≈ 𝐵 → (card‘𝐴) = (card‘𝐵))

Proof of Theorem carden2b

Step	Hyp	Ref	Expression
1		cardne 10034	. . . . 5 ⊢ ((card‘𝐵) ∈ (card‘𝐴) → ¬ (card‘𝐵) ≈ 𝐴)
2		ennum 10016	. . . . . . . 8 ⊢ (𝐴 ≈ 𝐵 → (𝐴 ∈ dom card ↔ 𝐵 ∈ dom card))
3	2	biimpa 476	. . . . . . 7 ⊢ ((𝐴 ≈ 𝐵 ∧ 𝐴 ∈ dom card) → 𝐵 ∈ dom card)
4		cardid2 10022	. . . . . . 7 ⊢ (𝐵 ∈ dom card → (card‘𝐵) ≈ 𝐵)
5	3, 4	syl 17	. . . . . 6 ⊢ ((𝐴 ≈ 𝐵 ∧ 𝐴 ∈ dom card) → (card‘𝐵) ≈ 𝐵)
6		ensym 9063	. . . . . . 7 ⊢ (𝐴 ≈ 𝐵 → 𝐵 ≈ 𝐴)
7	6	adantr 480	. . . . . 6 ⊢ ((𝐴 ≈ 𝐵 ∧ 𝐴 ∈ dom card) → 𝐵 ≈ 𝐴)
8		entr 9066	. . . . . 6 ⊢ (((card‘𝐵) ≈ 𝐵 ∧ 𝐵 ≈ 𝐴) → (card‘𝐵) ≈ 𝐴)
9	5, 7, 8	syl2anc 583	. . . . 5 ⊢ ((𝐴 ≈ 𝐵 ∧ 𝐴 ∈ dom card) → (card‘𝐵) ≈ 𝐴)
10	1, 9	nsyl3 138	. . . 4 ⊢ ((𝐴 ≈ 𝐵 ∧ 𝐴 ∈ dom card) → ¬ (card‘𝐵) ∈ (card‘𝐴))
11		cardon 10013	. . . . 5 ⊢ (card‘𝐴) ∈ On
12		cardon 10013	. . . . 5 ⊢ (card‘𝐵) ∈ On
13		ontri1 6429	. . . . 5 ⊢ (((card‘𝐴) ∈ On ∧ (card‘𝐵) ∈ On) → ((card‘𝐴) ⊆ (card‘𝐵) ↔ ¬ (card‘𝐵) ∈ (card‘𝐴)))
14	11, 12, 13	mp2an 691	. . . 4 ⊢ ((card‘𝐴) ⊆ (card‘𝐵) ↔ ¬ (card‘𝐵) ∈ (card‘𝐴))
15	10, 14	sylibr 234	. . 3 ⊢ ((𝐴 ≈ 𝐵 ∧ 𝐴 ∈ dom card) → (card‘𝐴) ⊆ (card‘𝐵))
16		cardne 10034	. . . . 5 ⊢ ((card‘𝐴) ∈ (card‘𝐵) → ¬ (card‘𝐴) ≈ 𝐵)
17		cardid2 10022	. . . . . 6 ⊢ (𝐴 ∈ dom card → (card‘𝐴) ≈ 𝐴)
18		id 22	. . . . . 6 ⊢ (𝐴 ≈ 𝐵 → 𝐴 ≈ 𝐵)
19		entr 9066	. . . . . 6 ⊢ (((card‘𝐴) ≈ 𝐴 ∧ 𝐴 ≈ 𝐵) → (card‘𝐴) ≈ 𝐵)
20	17, 18, 19	syl2anr 596	. . . . 5 ⊢ ((𝐴 ≈ 𝐵 ∧ 𝐴 ∈ dom card) → (card‘𝐴) ≈ 𝐵)
21	16, 20	nsyl3 138	. . . 4 ⊢ ((𝐴 ≈ 𝐵 ∧ 𝐴 ∈ dom card) → ¬ (card‘𝐴) ∈ (card‘𝐵))
22		ontri1 6429	. . . . 5 ⊢ (((card‘𝐵) ∈ On ∧ (card‘𝐴) ∈ On) → ((card‘𝐵) ⊆ (card‘𝐴) ↔ ¬ (card‘𝐴) ∈ (card‘𝐵)))
23	12, 11, 22	mp2an 691	. . . 4 ⊢ ((card‘𝐵) ⊆ (card‘𝐴) ↔ ¬ (card‘𝐴) ∈ (card‘𝐵))
24	21, 23	sylibr 234	. . 3 ⊢ ((𝐴 ≈ 𝐵 ∧ 𝐴 ∈ dom card) → (card‘𝐵) ⊆ (card‘𝐴))
25	15, 24	eqssd 4026	. 2 ⊢ ((𝐴 ≈ 𝐵 ∧ 𝐴 ∈ dom card) → (card‘𝐴) = (card‘𝐵))
26		ndmfv 6955	. . . 4 ⊢ (¬ 𝐴 ∈ dom card → (card‘𝐴) = ∅)
27	26	adantl 481	. . 3 ⊢ ((𝐴 ≈ 𝐵 ∧ ¬ 𝐴 ∈ dom card) → (card‘𝐴) = ∅)
28	2	notbid 318	. . . . 5 ⊢ (𝐴 ≈ 𝐵 → (¬ 𝐴 ∈ dom card ↔ ¬ 𝐵 ∈ dom card))
29	28	biimpa 476	. . . 4 ⊢ ((𝐴 ≈ 𝐵 ∧ ¬ 𝐴 ∈ dom card) → ¬ 𝐵 ∈ dom card)
30		ndmfv 6955	. . . 4 ⊢ (¬ 𝐵 ∈ dom card → (card‘𝐵) = ∅)
31	29, 30	syl 17	. . 3 ⊢ ((𝐴 ≈ 𝐵 ∧ ¬ 𝐴 ∈ dom card) → (card‘𝐵) = ∅)
32	27, 31	eqtr4d 2783	. 2 ⊢ ((𝐴 ≈ 𝐵 ∧ ¬ 𝐴 ∈ dom card) → (card‘𝐴) = (card‘𝐵))
33	25, 32	pm2.61dan 812	1 ⊢ (𝐴 ≈ 𝐵 → (card‘𝐴) = (card‘𝐵))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1537   ∈ wcel 2108   ⊆ wss 3976  ∅c0 4352   class class class wbr 5166  dom cdm 5700  Oncon0 6395  ‘cfv 6573   ≈ cen 9000  cardccrd 10004

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-sep 5317  ax-nul 5324  ax-pow 5383  ax-pr 5447  ax-un 7770

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3or 1088  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ne 2947  df-ral 3068  df-rex 3077  df-rab 3444  df-v 3490  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-pss 3996  df-nul 4353  df-if 4549  df-pw 4624  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-int 4971  df-br 5167  df-opab 5229  df-mpt 5250  df-tr 5284  df-id 5593  df-eprel 5599  df-po 5607  df-so 5608  df-fr 5652  df-we 5654  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-rn 5711  df-res 5712  df-ima 5713  df-ord 6398  df-on 6399  df-iota 6525  df-fun 6575  df-fn 6576  df-f 6577  df-f1 6578  df-fo 6579  df-f1o 6580  df-fv 6581  df-er 8763  df-en 9004  df-card 10008

This theorem is referenced by:  card1  10037  carddom2  10046  cardennn  10052  cardsucinf  10053  pm54.43lem  10069  nnadju  10267  nnadjuALT  10268  ficardun  10270  ackbij1lem5  10292  ackbij1lem8  10295  ackbij1lem9  10296  ackbij2lem2  10308  carden  10620  r1tskina  10851  cardfz  14021